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Essential cover and closure. (English) Zbl 1078.16015
If \(A\) is an essential two-sided ideal of the ring \(R\) (i.e. \(I\cap A\not=0\) for any nonzero ideal \(I\) of \(R\)), then \(R\) is called an essential extension of \(A\). By definition, the essential cover \(\mathcal{EM}\) of the class of rings \(\mathcal M\) consists of all essential extensions of rings belonging to \(\mathcal M\). The class \(\mathcal M\) is essentially closed if \(\mathcal{EM}={\mathcal M}\). The essential closure \({\mathcal M}^c\) is the union of the classes \({\mathcal M}^{(i)}\), where \({\mathcal M}^{(0)}={\mathcal M}\) and \({\mathcal M}^{(i+1)}=\mathcal{EM}^{(i)}\). The notions of essential cover and essential closure have been used successfully in the characterization of supernilpotent and special radicals.
In the paper under review the author considers rings with trivial two-sided annihilators and studies the problem for the characterization of the classes \(\mathcal M\) with the property that \({\mathcal M}^c={\mathcal M}^{(n+1)}\not={\mathcal M}^{(n)}\) for a given \(n\) and the classes \(\mathcal M\) such that \({\mathcal M}^{(n+1)}\not={\mathcal M}^{(n)}\) for all \(n\). He gives new examples showing that the construction of G. A. P. Heyman and C. Roos [J. Aust. Math. Soc., Ser. A 23, 340-347 (1977; Zbl 0375.16008)] of the essential closure can terminate at any finite place or at the first infinite ordinal.
MSC:
16N80 General radicals and associative rings
16D25 Ideals in associative algebras
16S70 Extensions of associative rings by ideals
16N60 Prime and semiprime associative rings
13G05 Integral domains
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